Equation of Straight Line in Plane/General Equation

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Theorem

A straight line $\mathcal L$ is the set of all $\tuple {x, y} \in \R^2$, where:

$\alpha_1 x + \alpha_2 y = \beta$

where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero.


Proof

Let $y = \map f x$ be the equation of a straight line $\mathcal L$.

From Line in Plane is Straight iff Gradient is Constant, $\mathcal L$ has constant slope.

Thus the derivative of $y$ with respect to $x$ will be of the form:

$y' = c$

Thus:

\(\displaystyle y\) \(=\) \(\displaystyle \int c \rd x\) Fundamental Theorem of Calculus
\(\displaystyle \) \(=\) \(\displaystyle c x + K\) Primitive of Constant

where $K$ is arbitrary.

Taking the equation:

$\alpha_1 x + \alpha_2 y = \beta$

it can be seen that this can be expressed as:

$y = - \dfrac {\alpha_1} {\alpha_2} x + \dfrac {\beta} {\alpha_2}$

thus demonstrating that $\alpha_1 x + \alpha_2 y = \beta$ is of the form $y = c x + K$ for some $c, K \in \R$.

$\blacksquare$


Sources